- Lecture 0 (01/08)
Definitions: Categories. Morphisms - injective, surjective, bijective,
left/right inverse, isomorphisms. Objects - sub/quotient, initial/final, null.
- Lecture 1 (01/10)
Functors. Natural transformations. Equivalence of categories.
- Lecture 2 (01/12)
Equivalence of categories cntd. Dual categories. Product of catgories.
There is a mistake in Lecture 2. See section 3.0 of the next lecture
- Lecture 3 (01/17)
Adjoint functors. Unit and counit of adjunction.
- Lecture 4 (01/19)
Adjoint functors cntd. Yoneda's lemma. Representable functors.
- Lecture 5 (01/22)
(Functors defining objects). Direct sums and direct products.
- Lecture 6 (01/24)
(Functors defining objects). Inverse limit.
- Lecture 7 (01/26)
Direct and inverse limits cntd.
- Lecture 8 (01/29)
Additive categories. Finite direct sums/products. Kernel and cokernel.
- Lecture 9 (01/31)
Kernel and cokernel continued. Abelian categories. Additive functors.
- Lecture 10 (02/02)
Notion of exactness: of sequences; of functors. Left/right exact functors.
- Lecture 26 (03/19)
Field extensions. Degree of an extension. Algebraic vs transcendental elements.
- Lecture 27 (03/21)
A little review. Splitting field of a polynomial - existence.
- Lecture 28 (03/23)
Splitting field - uniqueness.
Independence of group characters. Lower bound on degree of an extension.
- Lecture 29 (03/26)
Artin's theorem. Galois extensions. Normal and separable extensions.
Galois iff normal and separable (finite case).
- Lecture 30 (03/28)
Splitting field of a set of polynomials - existence and uniqueness.
Algebraically closed fields.
- Lecture 31 (03/30)
Some properties of normal extensions. Galois iff normal and separable
- Lecture 32 (04/02)
Fundamental theorem of Galois theory (finite case).
- Lecture 33 (04/04)
Differential criterion of separability. Finite fields. Perfect/imperfect
- Lecture 34 (04/06)
Primitive element theorem I.
- Lecture 35 (04/09)
Primitive element theorem II. Normal basis of Galois extensions.
- Lecture 36 (04/11)
Roots of unity. Noether's equations. A little bit of Galois cohomology.
- Lecture 37 (04/13)
Errata. Corollary on page 7 should have the condition
that sigma is not identity.
- Lecture 38 (04/16)
Norm, trace, Hilbert 90 and application to cyclic extensions.
- Lecture 39 (04/18)
Topological groups. Profinite groups.
- Lecture 40 (04/20)
Fundamental theorem of (infinite) Galois extensions.
Optional reading material
- Appendix A
Grothendieck's axioms (AB3-5) and proof of Baer's criterion of